Eur. Phys. J. D (2013) 67: 270DOI: 10.1140/epjd/e2013-40510-x

Regular Article

THE EUROPEANPHYSICAL JOURNAL D

An experimental investigation of the trap-dynamics of a cesiummagneto-optical trap at high laser intensities

Muhammad Anwara, Muhammad Faisal, and Mushtaq Ahmed

National Institute of Lasers and Optronics, P.O. Nilore, 44000 Islamabad, Pakistan

Received 18 August 2013 / Received in final form 8 October 2013Published online 23 December 2013 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2013

Abstract. We have developed Pakistan’s first magneto optical trap (MOT) in which we were able to captureas many as 2.8× 109 Cs atoms and attain densities as high as 2× 1011 atoms/cm3. The trap performancewas optimized by varying the laser intensity from 50 mW/cm2 to 500 mW/cm2 as well as the magneticfield gradient from 5 G/cm to 20 G/cm. We have fully investigated the dynamics of the cesium vapor cellmagneto-optical trap (VCMOT) by determining the capture rate L (atoms s−1), background collisional lossrate γ (s−1) and intra-trap excited-state collisional loss parameter β (cm3 s−1) of the cesium atoms in thetrap. Also, the cloud expansion and compression with trap laser intensity and magnetic field gradient wasobserved. The MOT dynamics at high intensities is characterized by high capture rate (4 × 109 atoms/s),high densities, enhanced collisional losses (β ∼ 10−11 cm3 s−1) and cloud expansion due to radiationtrapping. The intensities used in this study are the highest ever reported with the cesium MOT.

1 Introduction

During the resonant absorption (emission) of a photonby an atom, there is a complete transfer of photon mo-mentum (recoil) to the atom which can alter the internalas well as external degrees of freedom [1–3] of the atom.The 1-dimensional momentum exchange between photonsand atoms was observed in the experiments of decelera-tion [4], collimation [5,6] and deflection [7,8] of an atomicbeam by a resonant laser beam. The idea of 3-dimensionallaser cooling and trapping of atoms [9–11], was material-ized with the first observations of a optical molasses [12]and a optical dipole trap [13]. However, what is now knownas magneto-optical trap (MOT) was observed for the firsttime by Raab et.al. [14]. The first vapor cell magneto-optical trap (VCMOT) was with cesium [15] and laterwith sodium [16] atoms.

The magneto-optical trap dynamics has been exten-sively studied [13,17,18] during the last two decades andhave brought a revolution in the atomic and molecularphysics. The fountain atomic clocks with an accuracy ofone part in 1016 s are being used to study the varia-tion of universal constants with time; the Bose-Einsteincondensation (BEC) for producing macroscopic samplesof quantum matter; atom lasers, the atom-analog of laser,deliver beams of coherent matter; atom interferometers forprecision measurements and ultra-cold Rydberg atoms arethe new areas of research which emerged only because ofmagneto-optical traps [19–21].

a e-mail: mamalik2000@gmail.com

In the present work, we have developed and fullycharacterized a cesium vapor cell magneto-optical trap(VCMOT) similar to Monroe et al. [15]. We have inves-tigated the trap- dynamics at high laser intensities upto500 mW/cm2 never reported before with cesium MOT.We have carried out a systematic study of the capturerate, axial size, density and collisional losses as a functionof trap laser intensity and magnetic field gradient, keepingthe detuning fixed at –16 MHz. This experience is criticalin planning an advanced experiment with cold atoms inthe field of atomic and quantum-gas physics such as thestudy of ultracold Rydberg atoms, BEC and single atomquantum bit (QUBIT).

2 Experimental setup

A cesium magneto-optical trap was developed in the retro-reflection configuration using only three trapping beams.The three beams were then retro-reflected to produce atotal of six laser beams. Figure 1a shows detailed opticallayout of the experimental setup.

The trapping laser was a locally developed extendedcavity diode laser (ECDL) system consisting of a laserdiode (Model No. SDL-5412-H1), emitting at a wavelengthof 852 nm, 100 mW of optical power and an 1800 lines/mmdiffraction grating for extending the cavity. A pair ofanamorphic prisms was used for beam shaping. A –60 dBoptical isolation was provided to prevent any reflectedlight from reentering into the laser cavity. A fraction of

Page 2 of 10 Eur. Phys. J. D (2013) 67: 270

(a)

(b)

(c)

Fig. 1. Experimental setup of cesium magneto-optical trap(a) Complete Optical Layout showing lasers and MOT beams(b) Saturation Absorption Spectroscopy Setup for locking thetrapping laser to hyperfine level of cesium D2 line. (c) Fre-quency schematics of the trapping laser: CO (F ′ = 4, 5) is thecross-over of the two atomic levels, 6P3/2; F ′ = 4 and 6P3/2;F ′ = 5. Here ωo is the optical frequency of the trap laser pho-ton red-detuned by Δ = −16 MHz from the trapping transition6S1/2; F = 4 to 6P3/2; F ′ = 5 and ωs is the rf-frequency of theaccousto-optic modulator (AOM).

the trapping beam was split up for saturation absorptionspectroscopy (SAS) in a sealed cesium vapor cell. The laserhas to be protected against thermal and acoustic noise forfrequency stabilized operation.

The optical setup of the saturation absorption spec-troscopy for trapping laser consists of a convex lens (f =30 cm), an accousto-optical modulator (AOM), polarizingbeam-splitter cube (PBS) and a photodiode (PD). Thelaser beam is focused by the lens into the AOM for fre-quency shifting of the trapping beam. The AOM is drivenby a voltage-controlled oscillator @120 MHz ± 20 MHz.The AOM is used for red-detuning the trapping laserfrom the transition 62S1/2, F = 4 → 62P3/2, F ′ = 5 byΔ = −16 MHz. This is accomplished by down-shifting

the frequency ωo of the trapping beam by 109.5 MHzand locking it to the cross-over (CO) transition 62S1/2,F = 4 → 62P3/2, F ′ = CO(4,5) so that the trappingbeams (with frequency ωo) entering the MOT cell become109.5 MHz above the cross-over 62P3/2, F ′ = CO(4,5)level and 16 MHz below the 62P3/2, F ′ = 5 level, as shownin Figure 1c. Therefore, the trap laser is now detuned by–16 MHz from the transition 62S1/2, F = 4 to 62P3/2,F ′ = 5.

The trapping beam is then injected into a 250 mWMOPA (Master-Oscillator-Power-Amplifier). A half-waveplate (HWP) is used to rotate the polarization of thelinearly polarized trapping beam to “match” the crystalfacets of tapered amplifier. In order to ensure injectionlocking to the master laser, the power of the MOPA ismaximized and the ASE (amplified spontaneous emission)is minimized. The output of the MOPA is then passedthrough a large aperture optical isolator for protectionof the crystal against any reflected light from the MOTsetup.

The repumper laser is also a locally developed ex-tended cavity diode laser (ECDL) system comprising alaser diode emitting @ 852 nm, 50 mW of optical power, adiffraction grating and a pair of anamorphic prisms. Thelaser is provided –30 dB optical isolation against any re-flected light coming from MOT setup. The SAS setup forrepumper laser is similar to that of the trapping laser,without using any AOM, because the repumper laser islocked exactly to the transition 62S1/2, F = 3 → 62P3/2,F ′ = 4.

The re-pumper beam was combined with the trap-ping beam at a polarizing beamsplitter so that both laserbeams propagate together towards the MOT cell. Thecombined beams were then passed through a Galileantelescope of magnification 2X by using f1 = 75 mm andf2 = 150 mm for doubling the beams diameters. The finaldiameter of the beams was, however, controlled by insert-ing an iris. Also, it helped in the fine alignment of MOT.

The three equal-power trapping beams were producedby using a pair of the “half-wave plate (HWP)+ PBS”combination and arranged in retro-reflection configurationfor X , Y , and Z directions. A pair of quarter wave plates(QWP) was used in each of the three beams for ensuringσ+-σ− configuration for MOT operation.

The cesium vapor was obtained directly from the roomtemperature cesium background, provided by cesium ma-terial deposited on the walls of the MOT cell, at a vacuumof 2 × 10−9 mbar maintained by an 8 L/s ion pump. TheMOT cell is a pyrex-glass cell with two optical windowsalong vertical and eight optical windows in the horizontalplane.

Two coils of radius 50 mm were used in the anti-Helmholtz configuration with equal and opposite currents.The coils were designed to produce a magnetic field gra-dient of 10 G/cm at the center of the trap at a current of1.0 A and thereby Zeeman-shift the level 62P3/2, F ′ = 5,mF = ±5 of cesium atoms equal to the natural line-widthat the boundary of the trapping region.

Eur. Phys. J. D (2013) 67: 270 Page 3 of 10

3 MOT dynamics

The number of cesium atoms trapped in a magneto-opticaltrap can be described by the rate equation,

dNCs

dt= L − γNCs − β

∫V

n2Cs

(R, t) d3R. (1)

Here, the 1st term on R.H.S. is the capture or loadingrate of the magneto-optical trap whereas the 2nd and 3rdterms respectively give the background and volumetricloss rates. L, γ and β are the corresponding coefficientswhich completely determine the dynamics of the magneto-optical trap. Study of MOT dynamics consists in the mea-surement of these coefficients by an analysis of the MOTloading curve.

Let us assume that the MOT is operating in a regimewhere volumetric loss rate is negligible, then the dynami-cal equation of MOT becomes very simple,

dNCs

dt= L − γNCs , (2)

which has the solutions of the form,

N (t) = Nss(1 − e−γt). (3)

This is the usual MOT-loading equation, where Nss is thesteady state number of trapped atoms and γ = 1/τ is theloading time constant of the MOT.

3.1 The MOT loading curves: Nss and γ

The loading curves, which give the temporal evolution ofMOT, were recorded by collecting the fluorescence emittedby the trapped atoms by using a lens of diameter Φ 25 mmand a focal length f = 100 mm, placed at a distance of85 mm from the MOT. The lens focuses the fluorescenceonto the sensor of a calibrated PIN photodiode (ModelPIN-10DP-1) placed 25 mm behind the lens. Figure 2shows a typical loading curve of cesium MOT. The bestfit to this curve is, N (t) = Nss(1− e−γt). For this partic-ular loading curve, γ = 4.4 s−1 and Nss = 8.5 × 108 ce-sium atoms.

3.2 MOT size and density

An ordinary monochrome CCTV camera, with Si chip asits sensor, was used to capture the images of a cesiumMOT. Figure 3 shows one of the captured images of thecesium MOT. In order to estimate the MOT volume, weneed to calibrate our imaging system.

The shape of the MOT cloud is almost spherical whenthe number of atoms is ∼106 or less. As the trap laserintensity increases, the cloud shape starts deforming dueto beams imbalance and misalignment. Also, as the num-ber of atoms increases, the MOT expands, sometimesanisotropically, due to repulsive photon interactions. But

Fig. 2. Loading curve of a cesium MOT Trap laser intensity =234 mW/cm2; Magnetic field gradient = 15 G/cm and detun-ing of –16 MHz.

Fig. 3. CCD image of a cesium MOT; image magnification is1:1.

under the most common conditions, the MOT shape canbe approximated by an ellipsoid with volume 4

3πabc, wherea, b, and c are its semi-minor axes or ‘radii’ of theellipsoid.

In this study, the MOT was operating in the multi-ple scattering (MS) regime. In order to find its volume,we have to make two assumptions, that the shape of thecloud is an ellipsoid and that this cloud has a Gaussiandistribution of atoms along the three axes. Therefore, theGaussian radii of the MOT image shown in Figure 3b werea = 2.0 mm; b = 0.8 mm; c = 1.4 mm, so that its volumeis 5.28×10−3 cm3 and the corresponding MOT density asnCs = 1.61 × 1011 atoms/cm3. Usually, the MOT densitylies in the range of 1010 to 1011 atoms/cm3. This is the up-per limit of the normal MOT due to long range repulsiveinteractions in the MOT [22]. Higher densities can only beachieved in the dark-SPOT MOT [23].

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4 Results and discussion

4.1 Capture rate or Loading (L)

The capture rate or loading L is defined as

L =(

dN

dt

)t=0, N=0

. (4)

Applying this condition to the loading equation (3), weget

L = γNss (5)

where, Nss is the steady state number of atoms trappedin the MOT.

Therefore, we can find the capture rate or loading ei-ther by finding the slope of the loading curve at t = 0 orby using equation (5). We measured the capture rate orloading L at fixed trap laser detuning of Δ = −16 MHz.

4.1.1 Trap laser intensity dependence (Itrap)

We studied the intensity dependence of the capture rateL at a magnetic field gradient of 15 G/cm and the exper-imental results are shown in Figure 4a. As the trap laserintensity is increased, the loading also increases, reachesa maximum value for some optimum intensity and thenstarts decreasing at higher intensities. The atom loadingis the maximum at an intensity of 250 mW/cm2. It meansthat the capture of MOT is a maximum at this optimumintensity.

As the trap laser intensity is increased, the photonscattering rate by the atoms increases thereby increas-ing the scattering force or the spontaneous force on theatom Fsp = �kγp, where γp is the photon scattering rateand �k is the photon momentum. At sufficiently high in-tensities, γp approaches saturation to its maximum valueof γ/2 and corresponds to the maximum scattering forceof �kγ/2 on the atom. Therefore, at high intensities, dueto large force, the atoms with relatively large velocitiescan be quickly cooled and trapped thereby leading to highloading or capture rates. However, the decrease in the load-ing or capture rate with intensity is attributed to the de-crease in the damping coefficient β at high laser intensi-ties. The damping coefficient versus saturation parameterfor a single atom is shown in Figure 4c. It is clear thatthe damping coefficient for a single atom is maximum atso = 5, which corresponds to a quite low intensity. How-ever, for a large number of atoms in the background at adensity of 2× 108cm−3, the maximum in the damping co-efficient will occur at much high intensities with so ≥ 100.The maximum in the damping coefficient, therefore, leadsto a maximum in the loading or capture rate at an inten-sity of about 250 mW/cm2.

4.1.2 Magnetic field gradient dependence

The dependence of loading on magnetic field gradient atfixed value of trap laser intensity of ∼290 mW/cm2 was

(a)

(b)

100

101

102

103

0

0.05

0.1

0.15

0.2

0.25

so

β (h

k2 )−

(c)

Fig. 4. Loading and Damping Coefficient of a cesium MOT.(a) Loading (s−1) versus Trap Laser Intensity (Experiment)at magnetic field gradient = 15 G/cm; detuning = −16 MHz.(b) Loading (s−1) versus magnetic field gradient (experiment)at trap laser intensity of 290 mW/cm2; Detuning = −16 MHz.(c) Damping coefficient (in the units of �k2) versus saturationparameter so = I/Is (Simulation); detuning = −16 MHz.

Eur. Phys. J. D (2013) 67: 270 Page 5 of 10

also studied experimentally and the results are shown inthe Figure 4b. The loading L increases monotonically withan increase in magnetic field gradient up to 20 G/cm.

As the magnetic field gradient is increased, both theabsolute values of the magnetic field B, as well as thecorresponding Zeeman shift ≡ μBB/� increase. There-fore, in order to minimize the “effective” detuning i.e.,Δeff = −Δ+kv+μBB/�, the Zeeman contribution can becompensated by large values for the Doppler term. Sincek is fixed for a given wavelength of the trapping laserphoton, therefore, the Doppler term can be increased byincreasing the velocity v of the incoming atom. Thus, largemagnetic field gradients allow the cooling and trapping offast background atoms. Hence, loading or the capture rateincreases with the magnetic field gradient. Alternatively,we can say that the capture velocity of the MOT increaseswith magnetic field gradient.

4.2 Average steady-state number of trappedatoms (Nss)

The steady state number of trapped atoms (Nss) in the Csmagneto optical trap was studied as a function of trappinglaser intensity and magnetic field gradient at a fixed de-tuning of Δ = −16 MHz.

4.2.1 Trap laser intensity dependence (Itrap)

The dependence of Nss upon Itrap was studied experimen-tally at 15 G/cm as shown in Figure 5a. Nss has an op-timal dependence upon Itrap, i.e. Nss first increases withthe increase in trap laser intensity, reaches a maximumvalue and then decreases with intensity at higher valuesof Itrap.

As the intensity is increased, the capture rate L andhence the capture velocity vc increases with intensity. Alarge value of capture velocity means that a large fractionof the background atoms, with velocities 0 ≤ v ≤ vc, arenow available for cooling and trapping in the MOT as,N (vc) = nV C

∫ vc

0 e−12βmv2

v2dv, where V is the volumeof the trapping region, n = 2× 108 cm−3 is the density ofthe background cesium atoms, C is some constant relatedwith normalization and β = 1/kBT . Therefore, as thetrap laser intensity is increased, the vcand the steady statenumber of atoms increase in the MOT until the capturerate attains its maximum value and then starts decreasingwith intensity for higher values. The capture velocity andhence the steady state number of trapped atoms Nss thendecrease with an increase in intensity.

4.2.2 Magnetic field gradient dependencess

With fixed detuning and trap laser intensity, the steadystate number of atoms Nss in the MOT increases almostlinearly with the gradient dB/dz as shown in Figure 5b.As discussed before, large magnetic field gradients allowfast atoms to be cooled and trapped at quite high load-ing rates. Therefore, the capture velocity of the MOT in-creases with the magnetic field gradient and hence corre-spondingly the steady state number of trapped atoms also

(a)

(b)

Fig. 5. Steady state number of trapped atoms in cesium MOT.(a) Nss versus trap laser intensity at magnetic field gradient =15 G/cm; detuning = −16 MHz. (b) Nss versus magnetic fieldgradient at trap laser intensity of 292 mW/cm2; detuning =−16MHz.

increases. Alternatively, it is the trap-depth of the MOTwhich increases with magnetic field gradient thereby in-creasing the total number of trapped atoms. Therefore,a deeper MOT can have a large steady state number ofatoms.

4.3 Axial cloud size (Δz)

The MOT size is a result of an inter-play between the mag-netic field and the multiple scattering of photons amongthe atoms. The magnetic field is responsible for the com-pression of MOT in position space whereas the trap laserintensity is meant for velocity damping of the backgroundatoms thereby confining the atoms in momentum space.Greater the magnetic field gradient, larger the spring con-stant of the trap. Higher the trap laser intensity, higherthe capture of the MOT. Therefore, increasing both these

Page 6 of 10 Eur. Phys. J. D (2013) 67: 270

parameters, one should be able to indefinitely capture andcompress the atoms in a magneto optical trap.

However, both these processes are limited by repul-sive nonlinear interactions between the cold atoms, whichcome into action in the MOT as soon as its density reachessome maximum threshold value. These nonlinear interac-tions are briefly discussed below:

(a) Intensity-dependent cold collisions: Initially, thetrapped atoms have only inelastic collisions withthe background atoms. However, as the number oftrapped atoms increases, the density of the trappedatoms also increases and the cold atoms start collidinginside MOT. These are intensity-dependent cold col-lisions and are responsible for light induced trap loss.These collisions are characterized by long range molec-ular potentials so that the cold atoms in the MOTstart interacting over distances much longer than theirde Broglie wavelength.

(b) Positive photon pressure: At high enough MOTdensities, there is finite probability that a sponta-neously emitted photon by a cold atom may be re-captured in the MOT before the photon leaves thetrap. These re-absorptions develop a positive photonpressure in the MOT due to which it expands. Thehigher the trap laser intensity, the more this repulsiveforce acting outward from the trap center and hencegreater the MOT size. When the MOT size becomesintensity-dependent, the trap is believed to operate inthe multiple scattering (MS) regime.

(c) Shadow effect: Also, there is a 3rd nonlinear opticalinteraction taking place inside MOT, usually known asthe shadow effect, which results from the attenuationof the trapping beam while entering in the MOT. Dueto this absorption, the atoms lying close to the centerof MOT see shadow of the atoms lying outwards nearthe edge of the trap i.e., lesser intensity of trap lasercan reach the centre of MOT as compared with theouter region of the MOT. Hence, there will be lesserrepulsive radiation pressure in the interior of MOTand therefore the net effect will be a compression ofcloud.

We carried out a detailed study of the axial MOT sizeas a function of trap laser intensity and magnetic fieldgradient, keeping the detuning fixed at –16 MHz. The axialMOT size is an important parameter directly related tothe spring constant of the trap [24], therefore, this studyis an important part of the MOT dynamics at high laserintensities.

4.3.1 Trap laser intensity dependence (Itrap)

The axial MOT size was studied as a function of traplaser intensity at magnetic field gradient of 15 G/cm. Theresults are shown in Figure 6a.

As the intensity is increased, the MOT is compresseddue to increased radiative forces. This compression is fol-lowed by an expansion at still higher intensities.

(a)

(b)

(c)

Fig. 6. Axial MOT size (Δz) of a cesium MOT. (a) AxialMOT size versus trap laser Intensity at magnetic field gradientof 15 G/cm; detuning = −16 MHz. (b) Axial MOT size versusmagnetic field gradient at trap laser intensity of 230 mW/cm2;detuning = −16 MHz. (c) Axial MOT size versus Nss; de-tuning = −16 MHz; trap laser intensity = 50 mW/cm2

to 500 mW/cm2 and magnetic field gradient = 5 G/cm to20 G/cm.

Eur. Phys. J. D (2013) 67: 270 Page 7 of 10

The MOT compression with an increase in intensity isa result of the stronger radiative forces. During the netcompression, the light forces compress the atomic clouduntil it reaches its minimum size and the maximum den-sity. After reaching the maximum density, the MOT startsexpanding with intensity due to the positive radiationpressure in the multiple scattering regime.

4.3.2 Magnetic field gradient dependence

For a fixed value of trap laser intensity and detuning,the axial MOT size was studied as a function of the ap-plied magnetic field gradient. The results are shown inFigure 6b.

As the magnetic field gradient is increased, at an inten-sity of 230 mW/cm2, the steady state number of trappedatoms increases. Since the MOT is already operating be-yond the maximum density point at such a high intensity,therefore, an increase in Nss results in an increase in itsvolume and its axial size. The axial MOT size and also itsspring constant increases monotonically with the magneticfield gradient in our studies.

4.3.3 Axial MOT size versus Nss law

We combined all our axial MOT size data for trap laser in-tensities from 50 mW/cm2 to 500 mW/cm2 and magneticfield gradient from 5 G/cm to 20 G/cm at a fixed detuningof –16 MHz and plotted against the corresponding steadystate number of atoms Nss and the resultant plot is shownin Figure 6c. It was found that, unlike the past results [22],the axial size of our cesium MOT follows a superpositionlaw AN1/2 + BN1/3, with A = 0.34067 ± 0.18662 andB = 0.41807 ± 0.25322. Physical significance of this lawlies in the fact that our MOT has an optical density suchthat a spontaneously emitted photon will be scattered atleast once or at most twice before leaving the trap.

4.4 Average density (nCs) of trapped atoms

The expansion of MOT with an increase in trap laser in-tensity indicates the positive photon pressure inside MOTand therefore its density suffers from an upper limit [21],nmax = cκ

2σL(σR−σL )Itrap, where c, κ and Itrap have usual

meanings and σL is the absorption cross-section of pho-tons from the laser field and σR is the re-absorption cross-sections of photons in the MOT. Any further attempt toincrease the density beyond nmax would simply fail ei-ther because of increase in MOT size or because of light-induced trap loss. Therefore, the average MOT density isalso an important dynamical parameter of the magneto-optical trap which is a signature of both the damping andrestoring parts of the MOT force FMOT = −κr − βv. Wehave investigated the average density nCs of the trappedcesium atoms in a MOT as a function of the trap laser in-tensity and magnetic field gradient, keeping the detuningfixed at –16 MHz.

4.4.1 Trap laser intensity dependence (Itrap)

Keeping the trap laser detuning fixed, the average densityof cesium MOT was studied as a function of trap laserintensity at 15 G/cm. The experimental results thus ob-tained are plotted in the graphs in Figure 7a.

The average density nCs has an optimal dependencewith trap laser intensity. The density starts from a lowervalue at lower intensity, quickly increases to a maximumvalue at an intensity 170 mW/cm2 and then decreaseslinearly with intensity until a minimum value of densityat an intensity of 400 mW/cm2 is reached. The densitystarts increasing again for still higher intensity.

As the trap laser intensity is increased, the steady statenumber of atoms in the trap increases simultaneously witha decrease in the axial MOT size thereby increasing thedensity of the trapped atoms to its maximum value at170 mW/cm2. As the intensity is increased further, thedensity decreases due to multiple scattering and the light-induced collisions. Incidentally, the average MOT densitynCs is a minimum at an intensity of 410 mW/cm2, wherethe intensity dependent trap-loss is a maximum.

4.4.2 Magnetic field gradient dependence

The effect of magnetic field gradient on density was stud-ied at the trap laser intensity of 290 mW/cm2 and a de-tuning of –16 MHz and the results are shown in Figure 7b.

As the magnetic field gradient is increased, the den-sity decreases. The density is high at 5 G/cm becausethe MOT shrinks to its minimum size due to the en-hanced light induced trap loss at such a high intensityof ∼290 mW/cm2. As the magnetic field is increased from5 G/cm to 10 G/cm, both the steady state number ofatoms and MOT size increase due to large capture, there-fore, the density is reduced to its minimum value. How-ever, when the gradient is further increased from 10 G/cmto 20 G/cm the density increases again. This increase indensity is because the light induced trap loss at an inten-sity of ∼290 mW/cm2 has been greatly modified by highmagnetic field gradient, although both the population andsize of the MOT have increased in this operating range.

4.5 Background collisional loss rate (γ)

The inverse of the time constant of the MOT loadingcurve i.e., γ = 1/τ , is a measure of the background colli-sional loss rate of the MOT and is therefore constant fora fixed background density of cesium atoms in the MOTvapor cell. For a standard vapor cell magneto-optical trap(VCMOT), γ ∼ 10 s−1 under usual operating conditions,therefore, the corresponding loading time of the MOT isapproximately 100 ms.

We have studied γ as a function of trap laser intensityand magnetic field gradient at a fixed value of detuning of–16 MHz, Figure 8.

Page 8 of 10 Eur. Phys. J. D (2013) 67: 270

(a)

(b)

Fig. 7. Average density (nCs) of a cesium MOT. (a) Averagedensity of a Cs MOT versus intensity at magnetic field gradientof 15 G/cm; detuning = −16 MHz. (b) Average density of aCs MOT versus magnetic field gradient at trap laser intensityof 290 mW/cm2; detuning = −16 MHz.

4.5.1 Trap laser intensity dependence (Itrap)

The experimental results for the intensity dependence ofγ at a magnetic field gradient of 15 G/cm are shown inFigure 8a. The values of γ at the minimum intensity of50 mW/cm2 must be very close to the constant valueof gamma (γo) which is determined from the backgrounddensity of cesium atoms. Taking into consideration all thefour values of magnetic field gradient, the average valueof γo is approximately 4 s−1.

Now as the trap laser intensity is increased, γ first de-creases and then starts increasing again with intensity.This increase of γ with intensity is linear. At high intensi-ties, however, the γ saturates and decreases with intensitybeyond the maximum.

The intensity dependence of γ has its origin in thelight-induced cold collisions taking place in a magnetooptical trap. The colliding partners participating in a

(a)

(b)

Fig. 8. Background Collisional Loss rate γ of a cesium MOT.(a) γ versus trap laser intensity at magnetic field gradientof 15 G/cm; detuning = −16 MHz. (b) γ versus magneticfield gradient at fixed trap laser intensity of 292 mW/cm2;detuning = −16 MHz.

cold collision are quite slow with speeds ∼10 cm/s in theDoppler cooling regime, therefore, the corresponding col-lision times are ∼10−9 s. Since the cold collision times arecomparable with excited state lifetimes, therefore, the col-liding atoms once excited may emit a photon during thecollision which can alter the collision dynamics by impart-ing recoil to the emitting atom. However, its effect on theMOT performance appears in the form of loss of atomsfrom the trap.

Since these collisions take place in the presence of aresonant light, these are known as excited state collisionsin contrast with the ground state collisions which takeplace in the darkness and in which both atoms are in theground state.

The effective γ contains total collisional losses from theMOT, with the background cesium atoms as well as with

Eur. Phys. J. D (2013) 67: 270 Page 9 of 10

neighboring cold cesium atoms in the trap. Therefore, wesplit the γ into two parts,

γ = γo + γ(I) (6)

where γo is due to the collisions of trapped cesium atomswith background cesium atoms and γ(I), which is theintensity-dependent part of γ, is due to the binary col-lisions between cold cesium atoms trapped in the MOT.

The γ(I) is related to the MOT density and excitedstate collisions cross-section β by the relation, γ(I) =β nCs, where, nCs is the density of trapped cesium atoms.For a cesium MOT, the typical values are γ(I) ∼ 1 s−1,nCs ∼ 1011 cm−3 and β ∼ 10−11 cm3 s−1. Study of coldcollisions in a magneto optical trap involves the measure-ment of β from the knowledge of γ and density nCs ofMOT.

Gallaghar-Pritchard and Sesko et al. [25,26] have dis-cussed the intensity dependence of γ(I) respectively insodium and cesium atoms and attributed these losses tothe two processes, namely, radiative escape (RE) and finestructure changing collisions (FSCC). While one of thetwo colliding atoms undergoes a spontaneous emission ina radiative escape, in the fine-structure (FS) changing col-lision, if it survives the RE, it will change its fine struc-ture state from P3/2 to P1/2 beyond the Condon point.In both cases, the colliding atoms receive enough energyto escape from the trap. RE and FS are the dominantloss mechanisms if γ(I) increase with intensity. However,the increase of γ(I) with a decrease of intensity, in thelow intensity, might be because of the hyperfine-changingcollisions (HCC) between ground state cesium atoms. Inthis process, two cesium atoms can acquire the velocitiesof 5 m/s or 10 m/s, depending upon whether one or both ofthe colliding atoms change their state from 6S1/2(F = 4)to 6S1/2(F = 3), or vice versa, and both atoms may leavethe trap.

The HCC collisions may lead to the escape of the col-liding atoms as well as to the optical pumping of one orboth atoms into some dark state, i.e. state in which atomsno longer belong to the absorption-emission cycles of thetrapping transition. In either case, MOT suffers from anintensity-dependent loss of one or two cesium atoms.

4.5.2 Magnetic field gradient dependence

With fixed values of trap laser detuning and intensity, theparameter γ for collisional losses was studied as a functionof applied magnetic field gradient of the cesium MOT andresults are shown in Figure 8b. The results show that thecollisional losses in a magneto-optical trap increase withmagnetic field gradient reaching a maximum and then de-crease for higher values of gradient.

As the magnetic field gradient is increased, boththe loading and loss of atoms from the MOT increase.Whereas the loading increases monotonically with the gra-dient, however, the loss decreases due to an increased trap-depth at high magnetic field gradients. A high magneticfield gradient overcomes the losses by re-capturing the es-caping atoms after cold collision has taken place.

5 Conclusion

We have reported a vapor cell magneto optical trapof cesium atoms which has captured a maximum of3 × 109 atoms at reasonably high density of 2 ×1011 atoms/cm3. We have observed that the MOT ex-pands with trap laser intensity. Also, it was found thateffective trap loss rate varies linearly with trap laser inten-sity and therefore is an indicator of the intensity depen-dent intra-trap collisions. We have measured sufficientlyhigh loading rates ∼4×109 atoms/s with the modest traplaser beam diameter of 6.6 mm. We have carried out adetailed experimental investigation of the dynamics of acesium MOT over a large range of trap laser intensitiesfrom 50 mW/cm2 to 500 mW/cm2 although the magneticfield gradient was varied over a relatively narrow rangefrom 5 G/cm to 20 G/cm and the detuning was keptfixed. In another study, the cesium MOT was operatedin the temperature-limited regime, i.e. its size remainedfixed with intensity, and where the effects of detuningwere noticeable with a resolution of 1 MHz. However, inthe present study, the MOT was operated only in the MSregime and its capture was investigated at high laser in-tensity. We have summarized our findings regarding theMOT capture in the following.

As the intensity is increased, we increase the cap-ture rate resulting in an over-all gain in Nss and den-sity nCs of the trapped atoms. This is the temperature-limited regime of MOT operation. The density continuesto increase and attains a maximum value nmax at someintensity Ip. Further increase of intensity >Ip results inradiation trapping in the cloud. The phenomenon of mul-tiple scattering of photons produces a repulsive radiationpressure within the cloud. Eventually, the MOT expandsand its density drops. A drop in density results in reducedintra-trap collisional losses thereby increasing Nss again.It shows that we might be able to trap an increasinglyhigh number of atoms by increasing the laser intensity.However, the limitation to this process comes from thepower broadening or AC Stark splitting at high intensitieswhich greatly reduces the damping coefficient of the trap.The power-broadening contribute to the effective atom-photon coupling in the trap thereby saturating the sin-gle atom scattering rate and hence the spontaneous trap-ping forces in the trap. Therefore, in order to be able totrap an increasingly high number of atoms in the trap,we should increase the laser power but limit the laser in-tensity by correspondingly increasing the laser beams di-ameter. Gibble et al. [27] were able to capture 3.6 × 1010

cesium atoms at a density of 3.6 × 1010 cm−3 by using40 mm diameter laser beams at a trapping intensity of25 mW/cm2 per beam. In comparison, we could capture amaximum of 3 × 109 atoms at a density of 2× 1011 cm−3

by using nominal laser beam diameter of 6.6 mm at a totalof trapping laser intensity of 250 mW/cm2. Therefore, ourresults are quite impressive when the MOT density andthe corresponding trap volumes are compared.

In short, we have measured all the three constants,namely, L, γ and β of the dynamical equation, equa-tion (1), of MOT. Therefore, we have completely

Page 10 of 10 Eur. Phys. J. D (2013) 67: 270

determined the dynamics of cesium magneto-optical trapat high laser intensities. Also, we have qualitatively re-lated these dynamical constants with two other constants,namely, the damping coefficient β and the spring constantκ. The rigorous characterization of the MOT is a foun-dation on which the future cold-atom and quantum-gasphysics program of Pakistan will be built.

We are thankful to Stella T. Muller, Aida Bebeachibuli, DanielV. Magalhaes, and Vanderlei S.Bagnato from IFSC, Univer-sity of Sao Paulo, SP, Brazil for their valuable help in settingup Pakistan’s first cold atoms laboratory at National Insti-tute of Lasers and Optronics (NILOP), P.O. Nilore, Islamabad.We are also thankful to Prof. John Weiner for reviewing themanuscript.

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